2,481 research outputs found

    A generalization of Voronoi's reduction theory and its application

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    We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke Math.

    Lattices associated with vector spaces over a finite field

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    AbstractLet V denote the n-dimensional row vector space over a finite field Fq, and fix a subspace W of dimension n-d. Let L(n,d)=P∪{V}, where P is the set of all the subspaces of V intersecting trivially with W. Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials

    The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

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    When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group GG form a compact space \Cal C(G). We analyse the structure of \Cal C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group HH. We prove that \Cal C(H) is a 6-dimensional space that is path--connected but not locally connected. The lattices in HH form a dense open subset \Cal L(H) \subset \Cal C(H) that is the disjoint union of an infinite sequence of pairwise--homeomorphic aspherical manifolds of dimension six, each a torus bundle over (S3∖T)×R(\bold S^3 \smallsetminus T) \times \bold R, where TT denotes a trefoil knot. The complement of \Cal L(H) in \Cal C(H) is also described explicitly. The subspace of \Cal C(H) consisting of subgroups that contain the centre Z(H)Z(H) is homeomorphic to the 4--sphere, and we prove that this is a weak retract of \Cal C(H).Comment: Minor edits. Final version. To appear in the Pacific Journal. 41 pages, no figure
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